Is a traveler who began fasting in his own country, then moved to another country where they started fasting a day later than him, obligated to continue fasting with them, thus fasting for thirty-one days?

The determining factor for beginning the fast is the country from which one departed, and for ending it, the country to which one arrived. If the total number of days fasted is 28, then one day must be made up (qada'), because a month is never less than 29 days. If one completes 30 days of fasting in the country to which one has traveled, and the people of that country still have one day of fasting remaining, then one must fast with them, break the fast when they break it, and pray Eid with them.